CSCD 300 Data Structures Donald Shell’s Sorting Algorithm

CSCD 300 Data StructuresDonald Shell’s Sorting Algorithm Originally developed by Bill Clark, modified by Tom Capaul and Tim Rolfe

Shell Sort - Introduction • More properly, Shell’s Sort • Created in 1959 by Donald Shell • Link to a local copy of the article: • Donald Shell, “A High-Speed Sorting Procedure”, Communications of the ACM Vol 2, No. 7 (July 1959), 30-32 • Originally Shell built his idea on top of Bubble Sort (link to article flowchart), but it has since been transported over to Insertion Sort.

Shell Sort -General Description • Essentially a segmented insertion sort • Divides an array into several smaller non-contiguous segments • The distance between successive elements in one segment is called a gap. • Each segment is sorted within itself using insertion sort. • Then resegment into larger segments (smaller gaps) and repeat sort. • Continue until only one segment (gap = 1) - final sort finishes array sorting.

Shell Sort -Background • General Theory: • Makes use of the intrinsic strengths of Insertion sort. Insertion sort is fastest when: • The array is nearly sorted. • The array contains only a small number of data items. • Shell sort works well because: • It always deals with a small number of elements. • Elements are moved a long way through array with each swap and this leaves it more nearly sorted.

Shell Sort - example Initial Segmenting Gap = 4 93 30 68 42 80 60 12 85 10 30 93 68 42 10 60 12 85 80

Shell Sort - example (2) Resegmenting Gap = 2 10 30 60 12 42 93 68 85 80 10 12 42 30 60 85 68 93 80

Shell Sort - example (3) Resegmenting Gap = 1 10 12 42 30 60 85 68 93 80 10 12 30 42 60 68 80 85 93

Gap Sequences for Shell Sort • The sequence h1, h2, h3,. . . , ht is a sequence of increasing integer values which will be used as a sequence (from right to left) of gap values. • Any sequence will work as long as it is increasing and h1 = 1. • For any gap value hk we have A[i] <= A[i + hk] • An array A for which this is true is hk sorted. • An array which is hk sorted and is then hk-1 sorted remains hk sorted.

Shell Sort - Ideal Gap Sequence • Although any increasing sequence will work ( if h1 = 1): • Best results are obtained when all values in the gap sequence are relatively prime (sequence does not share any divisors). • Obtaining a relatively prime sequence is often not practical in a program so practical solutions try to approximate relatively prime sequences.

Shell Sort - Practical Gap Sequences • Three possibilities presented: • 1) Shell's suggestion - first gap is N/2 - successive gaps are previous value divided by 2. • Odd gaps only - like Shell method except if division produces an even number add 1. • better performance than 1) since all odd values eliminates the factor 2. • 2.2 method - like Odd gaps method (add 1 to even division result) but use a divisor of 2.2 and truncate. • best performance of all - most nearly a relatively prime sequence.

Shell Sort - Added Gap Sequence • Donald Knuth, in his discussion of Shell’s Sort, recommended another sequence of gaps. • h0 = 1 • hj+1 = hj * 3 + 1 • Find the hj > n, then start with hj/3

Link to the Java program that generated the above data.

Shell Sort - Time Complexity • Time complexity: O(nr) with 1 < r < 2 • This is better than O(n2) but generally worse than O(n log2n).

Shellsort - Code public static void shellSort( Comparable[ ] theArray, int n ) { // shellSort: sort first n items in array theArray for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; } }

ShellSort -Trace (gap = 4) for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; } [0] [1] [2] [3] [4] [5] [6] [7] [8] 60 80 93 12 42 30 68 85 10 theArray n: 9 gap: 4 i: j:

CSCD 300 Data Structures Donald Shell’s Sorting Algorithm